As you work more and more with Drupal, you will likely begin to formulate a list of modules that you feel are most useful to you and want to include them in each site that you create. There may be as many opinions on which ones are the most indispensible as there are Drupal builders but these are my current can't-live-without choices. In basketball terminology, here are my starting 5:

Administration Menu
As of Drupal 7, a Toolbar module was included in core to display a convenient static administration menu at the top of the site. If you want to have a version of this menu that looks similar but has a dropdown functionality, disable the Toolbar module and enable Administration Menu and Administration Menu Toolbar style within the Administration section on the modules page.

Backup and Migrate
With this module, you can backup your database to a destination of your choice and quickly restore it from that location if and when necessary. If you are using this module to migrate a database from one site to another, the versions of this module on both sites must match.

Node Export
Having the capability to, for instance, create nodes on a development site then copy them over to a production site is vital. This module allows you to easily do just that.

Pathauto
This module allows Drupal to create URLs that have more of a descriptive name like about-us rather than just the word node with a slash followed by a number. It requires the Token module so install that one too.

Views
If you need to extract data from the database for the purpose of displaying it in a certain format on your site, this is the ideal module for doing it. Its user interface allows you to select specific fields, filter the data, choose a sort order, and so much more.

Out of this idea to have a set of contributed and/or custom modules that you kick off a site with came the concept of installation profiles. These can be constructed and selected from the installation profile page during the installation process as an alternative to the standard or minimal ones. They can handle many other tasks beyond simply including specific modules but the point is that they have been designed as a tool which can be customized to fit your needs when initiating a new site.

Evolving from installation profiles are distributions, which could be described as entire units of file structures and databases to get your site up and running fast, acting the way you want it to.

As you continue to work with Drupal and build your own websites, you will come up with your own list of favorite modules.